The only cases where formal proof of theorems seem to have had actual mathematical value is for theorems that require checking a very large number of case, so much so that no human can be fully certain that no mistakes were made. Some examples:

A good explanation of how this insane system came up is given at Video "History of Oxford University by Chris Day (2018)".

As if it weren't enough, there are also the 6 Halls: permanent private hall.

The colleges are controlled by its fellows, a small self-electing body of highly successful scholars, usually in the dozens per college number it seems. Each college also usually has different types of fellows, e.g. see he university college page: www.univ.ox.ac.uk/about/college-fellowships/ (archive)

The college system does has its merits though, as it instates a certain sense of Hogwarts "belonging" to a certain group, so it might help students get better support for their learning projects from older students, or through the tutoring system. Of course, all such "belonging" feelings are bad, the correct thing would be to make great online tutorials for all, and answer questions in the open. But oh well, humans are dumb.

The college you are in impacts the quality of your courses, because tutorials are per-college. As of 2023, Ciro Santilli spoke to some students of the Computer science course of the University of Oxford, and was told that in some cases where you don't have anyone who can give the tutorial, you instead get a "class", i.e. a P.h.D. student going through question sheets with no interaction in the C.S. department, rather than a deep interactive discussion over the college fire. How can this system be so broken, it is beyond belief

This functionality is somewhat related to fraternities and sororities in 2000's United States.

Intuitively: unordered container where all the values are unique, just like C++ std::set.

More precisely for set theory formalization of mathematics:

Mnemonic: in means into. So we are going into a codomain that is large enough so that we can have a different image for every input.

In this section we classify some functions by the type of inputs and outputs they take and produce.

Is the only atom that has a closed form solution, which allows for very good predictions, and gives awesome intuition about the orbitals in general.

It is arguably the most important solution of the Schrodinger equation.

In particular, it predicts:

The explicit solution can be written in terms of spherical harmonics.

This was the first generation of commercially successful radios.

It uses a crystal detector as its diode, which is a crucial element of the radio, thus its name.

They were superseded by transistor radios, which were much more reliable, portable and could amplify the signal received.

4 squares are sufficient by Lagrange's four-square theorem.

3 is not enough by Legendre's three-square theorem.

The subsets reachable with 2 and 3 squares are fully characterized by Legendre's three-square theorem and

Bibliography:

The order of a algebraic structure is just its cardinality.

Sometimes, especially in the case of structures with an infinite number of elements, it is often more convenient to talk in terms of some parameter that characterizes the structure, and that parameter is usually called the degree.